A General System of Functional Equations Deriving From Additive, Quadratic, Cubic, and Quartic Mappings

In the current study, we introduce a system of functional equations (FEs) deriving from the mixed type additive–quadratic and the mixed-type cubic–quartic FEs which describes a multimixed additive–quadratic–cubic–quartic mapping. We also characterize such mappings and in fact, we represent the general system of the mixed-type additive-quadratic and the mixed-type cubic-quartic FEs defining a multimixed additive–quadratic–cubic–quartic mapping to one unified equation. Under some mild conditions, a multimixed additive–quadratic–cubic–quartic mapping is a multi–additive–quadratic–cubic–quartic mapping. Moreover, we establish the Găvruţa and Hyers–Ulam–Rassias (H-U-R) stability of a multimixed additive–quadratic–cubic–quartic FE in the setting of Banach spaces. Furthermore, we give a nonstable example for a multi–additive–quadratic–cubic–quartic function on Rn in the singularity case. Eventually, a numerical example is given that shows the difference between an approximate multi–additive–quadratic–cubic–quartic function and a multi–additive–quadratic–cubic–quartic function.